1. (negation consistency) When $\neg$ is a unary connective construed as negation, describes any theory $T$ such that for no formula $\phi$ in the language of $T$ do both $\phi \in T$ and $\neg \phi \in T$ hold.

2. (Post consistency) When $T$ is a theory, describes any theory $T$ for which there exists some formula $\phi$ such that $\phi \notin T$, i.e., any theory that is not trivial.